Quantifiers and Operations on Modalities and Contexts.
Tom Costello, Anna Patterson · 1998
We can reason about theories, as well as in them. Many natural phenomena can be captured by theories, often by introducing a modality, true just of the theory. Rather than treat these phenomena as the sentences true in this modality, we can treat the entire theory or modality as an object. This point of view was proposed by McCarthy, who proposed calling these reified modalities contexts. Contexts, viewed as theories or modalities, have structural properties, and associated natural structural operations, such as union and intersection. These structural properties are most naturally captured by an algebra, reminiscent of dynamic algebra. This algebra has a set of operations that can be applied to contexts to form new contexts. We introduce an algebra, reminiscent of dynamic algebra, but which acts on modalities, not relations. This algebra allows the construction of modalities from simpler modalities. We use the natural correspondence between a modality and the und...